Limits, Continuity & Differentiability
Continuity — Exponential Form Limits
nta_pyq_2024_apr
Grade 12
Question:
Let $f:(0,\pi)\to\mathbb{R}$ be a function given by $f(x)=\begin{cases}\left(\dfrac{8}{7}\right)^{\frac{\tan8x}{\tan7x}}, & 0<x<\dfrac{\pi}{2}\\ a-8, & x=\dfrac{\pi}{2}\\ (1+|\cot x|)^{\frac{b}{a}|\tan x|}, & \dfrac{\pi}{2}<x<\pi\end{cases}$ where $a,b\in\mathbb{Z}$. If $f$ is continuous at $x=\dfrac{\pi}{2}$, then $a^2+b^2$ is equal to:
Step-by-Step Solution
Key Concept: LHL at $\pi/2$: $\lim_{x\to(\pi/2)^-}(8/7)^{\tan8x/\tan7x}=(8/7)^0=1$ (since $\tan8x/\tan7x\to0$ as $x\to\pi/2$). So $a-8=1\Rightarrow a=9$. RHL: $\lim_{x\to(\pi/2)^+}(1+|\cot x|)^{(b/a)|\tan x|}=e^{b/a}$. Set $=1\Rightarrow b/a=0\Rightarrow b=0$.
LHL$=1\Rightarrow a-8=1\Rightarrow a=9$. RHL$=e^{b/9}=1\Rightarrow b=0$. $a^2+b^2=81$.
Correct Answer: 81